Abstract
Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P, the aperture angle of x with respect to Q is defined as the angle of the cone that: (1) contains Q, (2) has apex at x and (3) has its two rays emanating from x tangent to Q. We present algorithms with complexities O(n log in), O(n + n log (m/n)) and O(n + m) for computing the maximum aperture angle with respect to Q when x is allowed to vary in P. To compute the minimum aperture angle we modify the latter algorithm obtaining an O(n + m) algorithm. Finally, we establish an Ω(n + n log(m/n)) time lower bound for the maximization problem and an Ω(m + n) bound for the minimization problem thereby proving the optimality of our algorithms.
Original language | English (US) |
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Pages (from-to) | 411-435 |
Number of pages | 25 |
Journal | Algorithmica (New York) |
Volume | 33 |
Issue number | 4 |
DOIs | |
State | Published - 2002 |
Keywords
- Algorithms
- Aperture angle
- Complexity
- Computational geometry
- Convexity
- Discrete optimization
- Robotics
- Unimodality
- Visibility
ASJC Scopus subject areas
- General Computer Science
- Computer Science Applications
- Applied Mathematics